Note on hereditary normality of product spaces

In 1964 K. Morita introduced the concept of P(m)-spaces which characterized the normality of products with any metrizable space of weight ⩽ m. Especially, a topological space X is a normal P(ℵ0)-space if and only if X × Y is normal for any separable metrizable space Y. Okuyama (1991) introduced a similar notion of weak P (ℵ0)-spaces which concerned the paracompactness of products with any K-analytic space. is note, we show the difference of P(ℵ0)-spaces and weak P(ℵ0)-spaces, and also characterize a topological space X such that X × Y is hereditarily normal for any hereditarily normal, weak P(ℵ0)-space Y.